WORST_CASE(Omega(n^1),O(n^2))
proof of input_qu8uXbwMHo.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 481 ms]
    (10) BOUNDS(1, n^2)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 9 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 130 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 143 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
            (32) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
   trunc(0) -> 0
   trunc(s(0)) -> 0
   trunc(s(s(x))) -> s(s(trunc(x)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), trunc(x), s(y)) [1]
   trunc(0) -> 0 [1]
   trunc(s(0)) -> 0 [1]
   trunc(s(s(x))) -> s(s(trunc(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), trunc(x), s(y)) [1]
   trunc(0) -> 0 [1]
   trunc(s(0)) -> 0 [1]
   trunc(s(s(x))) -> s(s(trunc(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> f
true :: true:false
gt :: s:0 -> s:0 -> true:false
trunc :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
false :: true:false

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_f

----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(true, x, y) -> f(gt(x, y), trunc(x), s(y)) [1]
   trunc(0) -> 0 [1]
   trunc(s(0)) -> 0 [1]
   trunc(s(s(x))) -> s(s(trunc(x))) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> null_f
true :: true:false
gt :: s:0 -> s:0 -> true:false
trunc :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
false :: true:false
null_f :: null_f

Rewrite Strategy: INNERMOST
----------------------------------------

(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 1
   0 => 0
   false => 0
   null_f => 0

----------------------------------------

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z', z'') -{ 1 }-> f(gt(x, y), trunc(x), 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   gt(z, z') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0
   gt(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0
   gt(z, z') -{ 1 }-> 0 :|: v >= 0, z' = v, z = 0
   trunc(z) -{ 1 }-> 0 :|: z = 0
   trunc(z) -{ 1 }-> 0 :|: z = 1 + 0
   trunc(z) -{ 1 }-> 1 + (1 + trunc(x)) :|: x >= 0, z = 1 + (1 + x)

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[trunc(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(f(V1, V, V2, Out),1,[gt(V4, V3, Ret0),trunc(V4, Ret1),f(Ret0, Ret1, 1 + V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(trunc(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(trunc(V1, Out),1,[],[Out = 0,V1 = 1]).
eq(trunc(V1, Out),1,[trunc(V5, Ret11)],[Out = 2 + Ret11,V5 >= 0,V1 = 2 + V5]).
eq(gt(V1, V, Out),1,[],[Out = 0,V6 >= 0,V = V6,V1 = 0]).
eq(gt(V1, V, Out),1,[],[Out = 1,V1 = 1 + V7,V = 0,V7 >= 0]).
eq(gt(V1, V, Out),1,[gt(V8, V9, Ret2)],[Out = Ret2,V9 >= 0,V = 1 + V9,V1 = 1 + V8,V8 >= 0]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(trunc(V1,Out),[V1],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [gt/3]
1. recursive  : [trunc/2]
2. recursive  : [f/4]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gt/3
1. SCC is partially evaluated into trunc/2
2. SCC is partially evaluated into f/4
3. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations gt/3 
* CE 11 is refined into CE [12] 
* CE 10 is refined into CE [13] 
* CE 9 is refined into CE [14] 


### Cost equations --> "Loop" of gt/3 
* CEs [13] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [12] --> Loop 12 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations trunc/2 
* CE 8 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 6 is refined into CE [17] 


### Cost equations --> "Loop" of trunc/2 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [15] --> Loop 15 

### Ranking functions of CR trunc(V1,Out) 
* RF of phase [15]: [V1-1]

#### Partial ranking functions of CR trunc(V1,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V1-1


### Specialization of cost equations f/4 
* CE 5 is refined into CE [18] 
* CE 4 is refined into CE [19,20,21,22,23,24,25,26,27] 


### Cost equations --> "Loop" of f/4 
* CEs [26] --> Loop 16 
* CEs [27] --> Loop 17 
* CEs [25] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [22] --> Loop 20 
* CEs [21] --> Loop 21 
* CEs [23] --> Loop 22 
* CEs [20] --> Loop 23 
* CEs [19] --> Loop 24 
* CEs [18] --> Loop 25 

### Ranking functions of CR f(V1,V,V2,Out) 
* RF of phase [16,17]: [V-V2]

#### Partial ranking functions of CR f(V1,V,V2,Out) 
* Partial RF of phase [16,17]:
  - RF of loop [16:1]:
    V-V2
  - RF of loop [17:1]:
    V-2
    V/2-V2/2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [28,29,30,31,32,33] 
* CE 2 is refined into CE [34,35,36,37] 
* CE 3 is refined into CE [38,39,40,41] 


### Cost equations --> "Loop" of start/3 
* CEs [36,37,41] --> Loop 26 
* CEs [40] --> Loop 27 
* CEs [28] --> Loop 28 
* CEs [39] --> Loop 29 
* CEs [29,30,31,32,33,35] --> Loop 30 
* CEs [34,38] --> Loop 31 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


Computing Bounds
=====================================

#### Cost of chains of gt(V1,V,Out):
* Chain [[12],11]: 1*it(12)+1
  Such that:it(12) =< V1

  with precondition: [Out=0,V1>=1,V>=V1] 

* Chain [[12],10]: 1*it(12)+1
  Such that:it(12) =< V

  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [11]: 1
  with precondition: [V1=0,Out=0,V>=0] 

* Chain [10]: 1
  with precondition: [V=0,Out=1,V1>=1] 


#### Cost of chains of trunc(V1,Out):
* Chain [[15],14]: 1*it(15)+1
  Such that:it(15) =< Out

  with precondition: [V1=Out,V1>=2] 

* Chain [[15],13]: 1*it(15)+1
  Such that:it(15) =< Out

  with precondition: [V1=Out+1,V1>=3] 

* Chain [14]: 1
  with precondition: [V1=0,Out=0] 

* Chain [13]: 1
  with precondition: [V1=1,Out=0] 


#### Cost of chains of f(V1,V,V2,Out):
* Chain [[16,17],25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+0
  Such that:aux(5) =< V
aux(6) =< V-V2
aux(7) =< V-V2+1
it(17) =< V/2-V2/2
it(17) =< aux(5)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(7)
it(17) =< aux(7)
aux(4) =< aux(5)
aux(3) =< aux(5)-1
s(9) =< it(16)*aux(5)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [[16,17],19,25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+3
  Such that:aux(6) =< V-V2
aux(9) =< V
aux(10) =< 2*V-V2
it(17) =< aux(9)
s(13) =< aux(10)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(10)
it(17) =< aux(10)
aux(4) =< aux(9)
aux(3) =< aux(9)-1
s(9) =< it(16)*aux(9)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [[16,17],18,25]: 3*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+3
  Such that:aux(6) =< V-V2
aux(12) =< V
aux(13) =< 2*V-V2
it(17) =< aux(12)
s(15) =< aux(13)
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(13)
it(17) =< aux(13)
aux(4) =< aux(12)
aux(3) =< aux(12)-1
s(9) =< it(16)*aux(12)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

  with precondition: [V1=1,Out=0,V>=3,V2>=1,V>=V2+1] 

* Chain [25]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [24,25]: 3
  with precondition: [V1=1,V=0,Out=0,V2>=0] 

* Chain [23,25]: 3
  with precondition: [V1=1,V=1,V2=0,Out=0] 

* Chain [23,24,25]: 6
  with precondition: [V1=1,V=1,V2=0,Out=0] 

* Chain [22,25]: 1*s(17)+3
  Such that:s(17) =< 1

  with precondition: [V1=1,V=1,Out=0,V2>=1] 

* Chain [21,[16,17],25]: 4*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+3
  Such that:it(17) =< V/2
aux(14) =< V
it(16) =< aux(14)
it(17) =< aux(14)
aux(4) =< aux(14)
aux(3) =< aux(14)-1
s(9) =< it(16)*aux(14)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [21,[16,17],19,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+1*s(18)+6
  Such that:aux(10) =< 2*V
aux(15) =< V
s(18) =< aux(15)
it(16) =< aux(15)
s(13) =< aux(10)
it(16) =< aux(10)
aux(4) =< aux(15)
aux(3) =< aux(15)-1
s(9) =< it(16)*aux(15)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [21,[16,17],18,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+1*s(18)+6
  Such that:aux(13) =< 2*V
aux(16) =< V
s(18) =< aux(16)
it(16) =< aux(16)
s(15) =< aux(13)
it(16) =< aux(13)
aux(4) =< aux(16)
aux(3) =< aux(16)-1
s(9) =< it(16)*aux(16)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [21,25]: 1*s(18)+3
  Such that:s(18) =< V

  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [20,[16,17],25]: 4*it(16)+3*it(17)+2*s(9)+1*s(11)+1*s(12)+3
  Such that:it(17) =< V/2
aux(17) =< V
it(16) =< aux(17)
it(17) =< aux(17)
aux(4) =< aux(17)
aux(3) =< aux(17)-1
s(9) =< it(16)*aux(17)
s(11) =< it(17)*aux(4)
s(12) =< it(17)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [20,[16,17],19,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(13)+1*s(19)+6
  Such that:aux(10) =< 2*V
aux(18) =< V
s(19) =< aux(18)
it(16) =< aux(18)
s(13) =< aux(10)
it(16) =< aux(10)
aux(4) =< aux(18)
aux(3) =< aux(18)-1
s(9) =< it(16)*aux(18)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [20,[16,17],18,25]: 6*it(16)+2*s(9)+1*s(11)+1*s(12)+2*s(15)+1*s(19)+6
  Such that:aux(13) =< 2*V
aux(19) =< V
s(19) =< aux(19)
it(16) =< aux(19)
s(15) =< aux(13)
it(16) =< aux(13)
aux(4) =< aux(19)
aux(3) =< aux(19)-1
s(9) =< it(16)*aux(19)
s(11) =< it(16)*aux(4)
s(12) =< it(16)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=4] 

* Chain [20,25]: 1*s(19)+3
  Such that:s(19) =< V

  with precondition: [V1=1,V2=0,Out=0,V>=3] 

* Chain [19,25]: 2*s(13)+3
  Such that:aux(8) =< V
s(13) =< aux(8)

  with precondition: [V1=1,Out=0,V>=2,V2>=V] 

* Chain [18,25]: 2*s(15)+3
  Such that:aux(11) =< V
s(15) =< aux(11)

  with precondition: [V1=1,Out=0,V>=3,V2>=V] 


#### Cost of chains of start(V1,V,V2):
* Chain [31]: 1
  with precondition: [V1=0] 

* Chain [30]: 1*s(114)+18*s(118)+6*s(119)+24*s(120)+8*s(121)+8*s(124)+4*s(125)+4*s(126)+4*s(127)+2*s(128)+2*s(129)+3*s(133)+3*s(137)+2*s(140)+1*s(141)+1*s(142)+6*s(143)+4*s(144)+6*s(145)+4*s(146)+2*s(147)+2*s(148)+6
  Such that:s(114) =< 1
s(135) =< V-V2
s(132) =< V-V2+1
s(116) =< 2*V
s(136) =< 2*V-V2
s(117) =< V/2
s(133) =< V/2-V2/2
aux(27) =< V
s(118) =< aux(27)
s(119) =< s(117)
s(120) =< aux(27)
s(121) =< s(116)
s(120) =< s(116)
s(122) =< aux(27)
s(123) =< aux(27)-1
s(124) =< s(120)*aux(27)
s(125) =< s(120)*s(122)
s(126) =< s(120)*s(123)
s(119) =< aux(27)
s(127) =< s(118)*aux(27)
s(128) =< s(119)*s(122)
s(129) =< s(119)*s(123)
s(133) =< aux(27)
s(137) =< s(135)
s(133) =< s(135)
s(137) =< s(132)
s(133) =< s(132)
s(140) =< s(137)*aux(27)
s(141) =< s(133)*s(122)
s(142) =< s(133)*s(123)
s(143) =< aux(27)
s(144) =< s(136)
s(145) =< s(135)
s(143) =< s(135)
s(145) =< s(136)
s(143) =< s(136)
s(146) =< s(145)*aux(27)
s(147) =< s(143)*s(122)
s(148) =< s(143)*s(123)

  with precondition: [V1=1] 

* Chain [29]: 1
  with precondition: [V=0,V1>=1] 

* Chain [28]: 3
  with precondition: [V1>=0,V>=0,V2>=0] 

* Chain [27]: 1*s(149)+1
  Such that:s(149) =< V1

  with precondition: [V1>=1,V>=V1] 

* Chain [26]: 2*s(150)+1*s(152)+1
  Such that:s(152) =< V
aux(28) =< V1
s(150) =< aux(28)

  with precondition: [V1>=2] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [31] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [30] with precondition: [V1=1] 
    - Upper bound: nat(V)*48+7+nat(V)*18*nat(V)+nat(V)*6*nat(nat(V)+ -1)+nat(V)*6*nat(V-V2)+nat(V/2-V2/2)*nat(V)+nat(V)*2*nat(V/2)+nat(V/2-V2/2)*nat(nat(V)+ -1)+nat(nat(V)+ -1)*2*nat(V/2)+nat(2*V)*8+nat(V-V2)*9+nat(2*V-V2)*4+nat(V/2-V2/2)*3+nat(V/2)*6 
    - Complexity: n^2 
* Chain [29] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [28] with precondition: [V1>=0,V>=0,V2>=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [27] with precondition: [V1>=1,V>=V1] 
    - Upper bound: V1+1 
    - Complexity: n 
* Chain [26] with precondition: [V1>=2] 
    - Upper bound: 2*V1+1+nat(V) 
    - Complexity: n 

### Maximum cost of start(V1,V,V2): max([max([2,nat(V)*48+6+nat(V)*18*nat(V)+nat(V)*6*nat(nat(V)+ -1)+nat(V)*6*nat(V-V2)+nat(V/2-V2/2)*nat(V)+nat(V)*2*nat(V/2)+nat(V/2-V2/2)*nat(nat(V)+ -1)+nat(nat(V)+ -1)*2*nat(V/2)+nat(2*V)*8+nat(V-V2)*9+nat(2*V-V2)*4+nat(V/2-V2/2)*3+nat(V/2)*6]),nat(V)+V1+V1])+1 
Asymptotic class: n^2 
* Total analysis performed in 487 ms.


----------------------------------------

(10)
BOUNDS(1, n^2)

----------------------------------------

(11) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
   trunc(0) -> 0
   trunc(s(0)) -> 0
   trunc(s(s(z0))) -> s(s(trunc(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)
Tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
   TRUNC(0) -> c2
   TRUNC(s(0)) -> c3
   TRUNC(s(s(z0))) -> c4(TRUNC(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))
S tuples:
   F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
   TRUNC(0) -> c2
   TRUNC(s(0)) -> c3
   TRUNC(s(s(z0))) -> c4(TRUNC(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))
K tuples:none
Defined Rule Symbols:   f_3, trunc_1, gt_2

Defined Pair Symbols:   F_3, TRUNC_1, GT_2

Compound Symbols:   c_2, c1_2, c2, c3, c4_1, c5, c6, c7_1


----------------------------------------

(13) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
   TRUNC(0) -> c2
   TRUNC(s(0)) -> c3
   TRUNC(s(s(z0))) -> c4(TRUNC(z0))
   GT(0, z0) -> c5
   GT(s(z0), 0) -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

The (relative) TRS S consists of the following rules:

   f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
   trunc(0) -> 0
   trunc(s(0)) -> 0
   trunc(s(s(z0))) -> s(s(trunc(z0)))
   gt(0, z0) -> false
   gt(s(z0), 0) -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(15) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
   F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
   TRUNC(0') -> c2
   TRUNC(s(0')) -> c3
   TRUNC(s(s(z0))) -> c4(TRUNC(z0))
   GT(0', z0) -> c5
   GT(s(z0), 0') -> c6
   GT(s(z0), s(z1)) -> c7(GT(z0, z1))

The (relative) TRS S consists of the following rules:

   f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
   trunc(0') -> 0'
   trunc(s(0')) -> 0'
   trunc(s(s(z0))) -> s(s(trunc(z0)))
   gt(0', z0) -> false
   gt(s(z0), 0') -> true
   gt(s(z0), s(z1)) -> gt(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4

----------------------------------------

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, gt, trunc, GT, TRUNC, f

They will be analysed ascendingly in the following order:
gt < F
trunc < F
GT < F
TRUNC < F
gt < f
trunc < f

----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
gt, F, trunc, GT, TRUNC, f

They will be analysed ascendingly in the following order:
gt < F
trunc < F
GT < F
TRUNC < F
gt < f
trunc < f

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)

Induction Base:
gt(gen_s:0'8_8(0), gen_s:0'8_8(0)) ->_R^Omega(0)
false

Induction Step:
gt(gen_s:0'8_8(+(n12_8, 1)), gen_s:0'8_8(+(n12_8, 1))) ->_R^Omega(0)
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) ->_IH
false

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Lemmas:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
trunc, F, GT, TRUNC, f

They will be analysed ascendingly in the following order:
trunc < F
GT < F
TRUNC < F
trunc < f

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
trunc(gen_s:0'8_8(*(2, n319_8))) -> gen_s:0'8_8(*(2, n319_8)), rt in Omega(0)

Induction Base:
trunc(gen_s:0'8_8(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
trunc(gen_s:0'8_8(*(2, +(n319_8, 1)))) ->_R^Omega(0)
s(s(trunc(gen_s:0'8_8(*(2, n319_8))))) ->_IH
s(s(gen_s:0'8_8(*(2, c320_8))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Lemmas:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)
trunc(gen_s:0'8_8(*(2, n319_8))) -> gen_s:0'8_8(*(2, n319_8)), rt in Omega(0)


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
GT, F, TRUNC, f

They will be analysed ascendingly in the following order:
GT < F
TRUNC < F

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GT(gen_s:0'8_8(n810_8), gen_s:0'8_8(n810_8)) -> gen_c5:c6:c79_8(n810_8), rt in Omega(1 + n810_8)

Induction Base:
GT(gen_s:0'8_8(0), gen_s:0'8_8(0)) ->_R^Omega(1)
c5

Induction Step:
GT(gen_s:0'8_8(+(n810_8, 1)), gen_s:0'8_8(+(n810_8, 1))) ->_R^Omega(1)
c7(GT(gen_s:0'8_8(n810_8), gen_s:0'8_8(n810_8))) ->_IH
c7(gen_c5:c6:c79_8(c811_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Complex Obligation (BEST)

----------------------------------------

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Lemmas:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)
trunc(gen_s:0'8_8(*(2, n319_8))) -> gen_s:0'8_8(*(2, n319_8)), rt in Omega(0)


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
GT, F, TRUNC, f

They will be analysed ascendingly in the following order:
GT < F
TRUNC < F

----------------------------------------

(28) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(29)
BOUNDS(n^1, INF)

----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Lemmas:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)
trunc(gen_s:0'8_8(*(2, n319_8))) -> gen_s:0'8_8(*(2, n319_8)), rt in Omega(0)
GT(gen_s:0'8_8(n810_8), gen_s:0'8_8(n810_8)) -> gen_c5:c6:c79_8(n810_8), rt in Omega(1 + n810_8)


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
TRUNC, F, f

They will be analysed ascendingly in the following order:
TRUNC < F

----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TRUNC(gen_s:0'8_8(*(2, n1421_8))) -> gen_c2:c3:c410_8(n1421_8), rt in Omega(1 + n1421_8)

Induction Base:
TRUNC(gen_s:0'8_8(*(2, 0))) ->_R^Omega(1)
c2

Induction Step:
TRUNC(gen_s:0'8_8(*(2, +(n1421_8, 1)))) ->_R^Omega(1)
c4(TRUNC(gen_s:0'8_8(*(2, n1421_8)))) ->_IH
c4(gen_c2:c3:c410_8(c1422_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
F(true, z0, z1) -> c(F(gt(z0, z1), trunc(z0), s(z1)), GT(z0, z1))
F(true, z0, z1) -> c1(F(gt(z0, z1), trunc(z0), s(z1)), TRUNC(z0))
TRUNC(0') -> c2
TRUNC(s(0')) -> c3
TRUNC(s(s(z0))) -> c4(TRUNC(z0))
GT(0', z0) -> c5
GT(s(z0), 0') -> c6
GT(s(z0), s(z1)) -> c7(GT(z0, z1))
f(true, z0, z1) -> f(gt(z0, z1), trunc(z0), s(z1))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(z0))) -> s(s(trunc(z0)))
gt(0', z0) -> false
gt(s(z0), 0') -> true
gt(s(z0), s(z1)) -> gt(z0, z1)

Types:
F :: true:false -> s:0' -> s:0' -> c:c1
true :: true:false
c :: c:c1 -> c5:c6:c7 -> c:c1
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
GT :: s:0' -> s:0' -> c5:c6:c7
c1 :: c:c1 -> c2:c3:c4 -> c:c1
TRUNC :: s:0' -> c2:c3:c4
0' :: s:0'
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6:c7
c6 :: c5:c6:c7
c7 :: c5:c6:c7 -> c5:c6:c7
f :: true:false -> s:0' -> s:0' -> f
false :: true:false
hole_c:c11_8 :: c:c1
hole_true:false2_8 :: true:false
hole_s:0'3_8 :: s:0'
hole_c5:c6:c74_8 :: c5:c6:c7
hole_c2:c3:c45_8 :: c2:c3:c4
hole_f6_8 :: f
gen_c:c17_8 :: Nat -> c:c1
gen_s:0'8_8 :: Nat -> s:0'
gen_c5:c6:c79_8 :: Nat -> c5:c6:c7
gen_c2:c3:c410_8 :: Nat -> c2:c3:c4


Lemmas:
gt(gen_s:0'8_8(n12_8), gen_s:0'8_8(n12_8)) -> false, rt in Omega(0)
trunc(gen_s:0'8_8(*(2, n319_8))) -> gen_s:0'8_8(*(2, n319_8)), rt in Omega(0)
GT(gen_s:0'8_8(n810_8), gen_s:0'8_8(n810_8)) -> gen_c5:c6:c79_8(n810_8), rt in Omega(1 + n810_8)
TRUNC(gen_s:0'8_8(*(2, n1421_8))) -> gen_c2:c3:c410_8(n1421_8), rt in Omega(1 + n1421_8)


Generator Equations:
gen_c:c17_8(0) <=> hole_c:c11_8
gen_c:c17_8(+(x, 1)) <=> c(gen_c:c17_8(x), c5)
gen_s:0'8_8(0) <=> 0'
gen_s:0'8_8(+(x, 1)) <=> s(gen_s:0'8_8(x))
gen_c5:c6:c79_8(0) <=> c5
gen_c5:c6:c79_8(+(x, 1)) <=> c7(gen_c5:c6:c79_8(x))
gen_c2:c3:c410_8(0) <=> c2
gen_c2:c3:c410_8(+(x, 1)) <=> c4(gen_c2:c3:c410_8(x))


The following defined symbols remain to be analysed:
F, f